A few remarks on a small example by Jean-Claude Deville regarding non-ignorable non-response Section 3. Estimation using the method of momentsA few remarks on a small example by Jean-Claude Deville regarding non-ignorable non-response Section 3. Estimation using the method of moments

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3.1 MAR

The method of moments makes it possible to estimate parameters quickly. For MAR, we obtain the third column of Table 2.3 using the equations

$$\begin{array}{ll}\text{E}\left(m_H\right)\hfill & \mathrm{<mo>=</mo>}{n}_{H\mathrm{.}}\left(1-p_H\right)\mathrm{,}\hfill \\ \text{E}\left(m_F\right)\hfill & \mathrm{<mo>=</mo>}{n}_{F\mathrm{.}}\left(1-p_F\right)\mathrm{,}\hfill \end{array}$$

which yield the estimators

$$\begin{array}{ll}{\widehat{p}}_H\hfill & \mathrm{<mo>=</mo>\; 1}-\frac{m_H}{n_{H\mathrm{.}}}\mathrm{,}\hfill \\ {\widehat{p}}_F\hfill & \mathrm{<mo>=</mo>\; 1}-\frac{m_F}{n_{F\mathrm{.}}}\mathrm{,}\hfill \end{array}$$

and therefore, from the first two columns,

$$\begin{array}{lll}{\widehat{n}}_{\mathrm{.}D}\hfill & \mathrm{<mo>=</mo>}\frac{r_{HD}}{{\widehat{p}}_H}+\frac{r_{FD}}{{\widehat{p}}_F}\hfill & \mathrm{<mo>=</mo>}{r}_{HD}\frac{n_{H\mathrm{.}}}{n_{H\mathrm{.}}-m_H}+r_{FD}\frac{n_{F\mathrm{.}}}{n_{F\mathrm{.}}-m_F}\mathrm{,}\hfill \\ {\widehat{n}}_{\mathrm{.}S}\hfill & \mathrm{<mo>=</mo>}\frac{r_{HS}}{{\widehat{p}}_H}+\frac{r_{FS}}{{\widehat{p}}_F}\hfill & \mathrm{<mo>=</mo>}{r}_{HS}\frac{n_{H\mathrm{.}}}{n_{H\mathrm{.}}-m_H}+r_{FS}\frac{n_{F\mathrm{.}}}{n_{F\mathrm{.}}-m_F}\mathrm{.}\hfill \end{array}$$

The estimated response probabilities are ${\widehat{p}}_H\mathrm{<mo>=</mo>}\text{0}\text{.4}$ and ${\widehat{p}}_F\mathrm{<mo>=</mo>}\text{0}\text{.6}.$ We therefore obtain the estimates shown in Table 3.1.

3.2 NMAR

For NMAR, we obtain the following equations from Table 2.4:

$$\begin{array}{ll}\text{E}\left(m_H\right)\hfill & \mathrm{<mo>=</mo>}E\left(r_{HD}\right)\frac{1-q_D}{q_D}+\text{E}\left(r_{HS}\right)\frac{1-q_S}{q_S}\mathrm{,}\hfill \\ \text{E}\left(m_F\right)\hfill & \mathrm{<mo>=</mo>}E\left(r_{FD}\right)\frac{1-q_D}{q_D}+\text{E}\left(r_{FS}\right)\frac{1-q_S}{q_S}\mathrm{.}\hfill \end{array}$$

After a few calculations, we obtain the following response probability estimators:

$$\begin{array}{ll}{\widehat{q}}_D\hfill & \mathrm{<mo>=</mo>}\frac{r_{HD}{r}_{FS}-r_{FD}{r}_{HS}}{\left(m_H+r_{HD}\right)r_{FS}-\left(m_F+r_{FD}\right)r_{HS}}\mathrm{,}\hfill \\ {\widehat{q}}_S\hfill & \mathrm{<mo>=</mo>}\frac{r_{HD}{r}_{FS}-r_{FD}{r}_{HS}}{\left(m_F+r_{FS}\right)r_{HD}-\left(m_H+r_{HS}\right)r_{FD}}\mathrm{.}\hfill \end{array}$$

Finally, we obtain

$$\begin{array}{llll}{\widehat{n}}_{\mathrm{.}D}\hfill & \mathrm{<mo>=</mo>}\frac{r_{\mathrm{.}D}}{{\widehat{q}}_D}\hfill & \mathrm{<mo>=</mo>}{r}_{\mathrm{.}D}\frac{\left(m_H+r_{HD}\right)r_{FS}-\left(m_F+r_{FD}\right)r_{HS}}{r_{HD}{r}_{FS}-r_{FD}{r}_{HS}}\hfill & \mathrm{<mo>=</mo>}{r}_{\mathrm{.}D}\frac{n_{H\mathrm{.}}{r}_{FS}-n_{F\mathrm{.}}{r}_{HS}}{r_{HD}{r}_{FS}-r_{FD}{r}_{HS}}\mathrm{,}\hfill \\ {\widehat{n}}_{\mathrm{.}S}\hfill & \mathrm{<mo>=</mo>}\frac{r_{\mathrm{.}S}}{{\widehat{q}}_S}\hfill & \mathrm{<mo>=</mo>}{r}_{\mathrm{.}S}\frac{\left(m_F+r_{FS}\right)r_{HD}-\left(m_H+r_{HS}\right)r_{FD}}{r_{HD}{r}_{FS}-r_{FD}{r}_{HS}}\hfill & \mathrm{<mo>=</mo>}{r}_{\mathrm{.}S}\frac{n_{F\mathrm{.}}{r}_{HD}-n_{H\mathrm{.}}{r}_{FD}}{r_{HD}{r}_{FS}-r_{FD}{r}_{HS}}\mathrm{.}\hfill \end{array}$$

As Deville writes, the estimated response probabilities are ${\widehat{q}}_D\mathrm{<mo>=</mo>}\text{0}\text{.2}$ and ${\widehat{q}}_S\mathrm{<mo>=</mo>}\text{0}\text{.8}.$ We therefore obtain the estimates in Table 3.2.

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